Elasticity of short DNA molecules: theory and experiment for contour lengths of 0.6-7 microm - PubMed (original) (raw)
Elasticity of short DNA molecules: theory and experiment for contour lengths of 0.6-7 microm
Yeonee Seol et al. Biophys J. 2007.
Abstract
The wormlike chain (WLC) model currently provides the best description of double-stranded DNA elasticity for micron-sized molecules. This theory requires two intrinsic material parameters-the contour length L and the persistence length p. We measured and then analyzed the elasticity of double-stranded DNA as a function of L (632 nm-7.03 microm) using the classic solution to the WLC model. When the elasticity data were analyzed using this solution, the resulting fitted value for the persistence length p(wlc) depended on L; even for moderately long DNA molecules (L = 1300 nm), this apparent persistence length was 10% smaller than its limiting value for long DNA. Because p is a material parameter, and cannot depend on length, we sought a new solution to the WLC model, which we call the "finite wormlike chain (FWLC)," to account for effects not considered in the classic solution. Specifically we accounted for the finite chain length, the chain-end boundary conditions, and the bead rotational fluctuations inherent in optical trapping assays where beads are used to apply the force. After incorporating these corrections, we used our FWLC solution to generate force-extension curves, and then fit those curves with the classic WLC solution, as done in the standard experimental analysis. These results qualitatively reproduced the apparent dependence of p(wlc) on L seen in experimental data when analyzed with the classic WLC solution. Directly fitting experimental data to the FWLC solution reduces the apparent dependence of p(fwlc) on L by a factor of 3. Thus, the FWLC solution provides a significantly improved theoretical framework in which to analyze single-molecule experiments over a broad range of experimentally accessible DNA lengths, including both short (a few hundred nanometers in contour length) and very long (microns in contour length) molecules.
Figures
FIGURE 1
Cartoon illustrating the experimental geometry (not to scale). A DNA molecule (green) is linked at one end to a surface. The opposite end of the DNA is attached to a bead held in an optical trap (orange). The trap center is offset a lateral distance _x_DNA from the surface attachment point. The vertical distance of the trap center from the surface is _z_trap. The bead displacement in both axes (_x_bd and _z_bd) is measured from the center of the trap (dashed line) to the center of the bead (solid line). The extension, x, of the DNA along the stretching axis can be determined from the geometry (24).
FIGURE 2
Sketch of idealized experimental geometries for single-molecule force experiments with nucleic-acid polymers. The total molecule extension is x. (A) Idealized pulling geometry in which force is applied directly to the ends of the molecule. (B) One-bead geometry. The molecule is attached at one end to a surface (glass slide or coverslip) and at the other end to a bead. Force is applied to the center of the bead. (C) Two-bead geometry. The molecule is attached at both ends to beads.
FIGURE 3
Variables used to describe chain conformations. Force is applied in the direction. The arc length along the chain is s, and the vector r(s) = (x(s), y(s), z(s)) is the coordinate of the chain at arc length s. The vector
is the unit vector tangent to the chain at s.
FIGURE 4
Sketch of the boundary conditions. (A) Unconstrained boundary conditions. The tangent vector at the end of the chain (short arrow) points in any direction relative to the direction of applied force (notched arrowhead). (B) Half-constrained boundary conditions. Due to a constraint (such as the presence of a solid surface), the tangent vector at the end of the chain can point only in the upper half-sphere. (C) Perpendicular boundary conditions. The tangent vector is constrained to point only normal to the surface in the direction of the applied force. (Note that boundary conditions are necessary for both ends (s = 0 and s = L) of the chain.) (D) Variables used to describe bead rotational fluctuations. The vector points from the center of the bead to the polymer-bead attachment. The angle θ between the
direction and the bead-chain attachment is defined by
The angle γ is defined by
where
is the tangent vector at the end of the chain. The bead has radius R.
FIGURE 5
Effective chain-end boundary conditions induced by bead rotational fluctuations. The unnormalized probability density is shown versus
for a bead of radius R = 250 nm. The different curves correspond to different values of the applied force. In this calculation, the attachment of the chain to the bead is half-constrained: in the absence of bead rotational fluctuations, the probability density is zero for cos γ < 0 and constant for cos _γ_ > 0. This step function is approached for a large force. For smaller values of FR, the probability density is smeared out and approaches a constant value (independent of cos γ).
FIGURE 6
Convergence of the FWLC-predicted extension to the classic WLC prediction for large L. The predicted fractional extension x/L is shown as a function of the contour length L (at fixed applied force). The different curves correspond to different theoretical assumptions: 1), the classic WLC solution, calculated using the method of Marko and Siggia (4); 2), the FWLC with unconstrained boundary conditions at both ends of the chain; 3), the FWLC with half-constrained boundary conditions at both ends of the chain; and 4), the FWLC with perpendicular boundary conditions at both ends of the chain (see Fig. 4). The different panels show different applied forces: (A) 0.1 pN, (B) 1 pN. The polymer persistence length is p = 50 nm. Note that the WLC prediction is independent of contour length, and the FWLC predictions converge to the WLC prediction as L increases. The convergence occurs more quickly for higher applied force (note different y axis scales for different panels). For low applied force, the more constrained boundary conditions lead to a larger predicted fractional extension (see text).
FIGURE 7
Results of fitting FWLC solution predictions to the classic WLC solution. The fitted value of the persistence length _p_∞ is shown as a function of contour length L. The polymer persistence length is p = 50 nm. (A) No bead. The different curves correspond to the unconstrained, half-constrained, and perpendicular boundary conditions at both ends of the chain. The apparent p decreases most for the half-constrained boundary conditions. The apparent persistence length decreases by >10 nm as L decreases from 105 to 100 nm. (B) One or two beads attached to the chain ends, with half-constrained boundary conditions at both ends of the chain.
FIGURE 8
Force-extension curves predicted by the FWLC solution for L = 500 nm (L/p = 10) and different boundary conditions. The different curves correspond to the classic WLC solution and the FWLC with unconstrained, half-constrained, and perpendicular boundary conditions at both ends of the chain. The predicted extension for all boundary conditions converges at high force. For a low applied force <80 fN (shaded), the more constrained boundary conditions lead to a larger predicted extension, though our model may not adequately address steric interactions at these forces (see text).
FIGURE 9
Experimental elasticity results. (A) Elasticity measurement of the 864-nm long DNA tether (magenta) stretched up to 8 pN in the positive and negative direction. A fit to the WLC solution (black) returns three values, _p_wlc, L, and Δ, where _p_wlc is the apparent persistence length, L is the contour length, and Δ is the location of the tether point. For the trace shown, the fitted values were 45.0 (±0.6) nm, −0.2 (±0.1) nm, and 863.0 (±0.4) nm, respectively. Data near the origin (1.2 R) are not used in the analysis due to experimental geometric constraints, where R is the radius of the trapped bead. (B) Force-versus-extension traces, with WLC fits, for a range of DNA lengths: 632 nm (lt. blue), 666 nm (lt. green), 756 nm (gold), 864 nm (magenta), 1007 nm (blue), 1514 nm (green), 2013 nm (orange), 2413 nm (purple), and 7030 nm (wine). Only the positive portion of the full records is shown for clarity. (C) Persistence length (mean ± SE) as a function of DNA length (top axis). The fit (green line) represents an empirical approximation to the experimental data over the range of DNA lengths studied where _p_wlc = _p_∞/(1 + a _p_∞/L), _p_∞ = 51.5 nm (dashed line) is the persistence length for an infinitely long DNA molecule, and a = 2.78. Extrapolation of this curve outside the experimentally tested length range is shown (dotted line). Since _p_wlc depends on L, we plot _p_wlc as a function of the number of persistence lengths (L/_p_∞) by using the length-independent _p_∞ (bottom axis).
FIGURE 10
Dependence of p_wlc on experimental parameters. (A) Percent variation in apparent persistence length (Δ_p/_p_0); mean ± SE) as a function of minimum force F_min used in fitting the data for two different DNA lengths, 666 nm (magenta) and 2413 nm (purple), where Δ_p = _p_0 – _p_F and _p_0 is the _p_wlc determined from fitting all of the data, and _p_F is the _p_wlc determined from the fit after excluding data below an absolute value _F_min in both the positive and negative directions. (B) Variation in persistence length (mean ± SE) as a function of the highest force (_F_max) used in fitting the data for two different DNA lengths, 666 nm (magenta) and 2413 nm (purple). A linear fit (dashed line) to the data yields a dependence of _p_wlc on _F_max of −1.8 nm/pN. (C) Persistence length (mean ± SE) as a function of DNA length at two different _F_max (6 and 8 pN) shows a systematic offset (as described in B). (D) The difference between the two traces in panel C shows a constant offset, suggesting the observed reduction of _p_wlc as a function of L is robust, although its absolute magnitude depends on _F_max.
FIGURE 11
Force-extension curves predicted by the FWLC solution for L = 500 nm (L/p = 10) with one bead. We implemented half-constrained boundary conditions at both ends of the chain but included bead rotational fluctuations. The predicted molecular extensions, x (after the bead radius has been subtracted), agree with the no-bead prediction for large force, but decrease more rapidly as the force decreases. Such a reduction in x becomes more significant for bigger sized beads. The curves for two different beads (R = 100 and 250 nm) are indistinguishable over the theoretically valid force range (80 fN < F < 8 pN). Below 80 fN (shaded), the predicted x for a bead with R = 250 nm bead decreases faster than x for a bead with R = 100 nm.
FIGURE 12
Comparison between finite and classic (infinite chain) WLC solutions. (A) Persistence length determined by fitting experimental data to the classic WLC solution (_p_wlc, red) and the finite WLC solution using half-constrained boundary conditions (_p_fin, blue) as a function of DNA length. (B) Fractional error in the fitted p from _p_∞ showing an approximately threefold reduction in the length dependence of _p_fin as compared to _p_wlc. Ideally, p, an intrinsic property of the molecule, would be independent of L. Error bars represent the standard error of the mean.
Comment in
- The great hunt for extra compliance.
Liphardt J. Liphardt J. Biophys J. 2007 Dec 15;93(12):4099. doi: 10.1529/biophysj.107.117572. Epub 2007 Aug 31. Biophys J. 2007. PMID: 17766362 Free PMC article. No abstract available.
References
- Smith, S. B., L. Finzi, and C. Bustamante. 1992. Mechanical measurements of the elasticity of single DNA molecules by using magnetic beads. Science. 258:1122–1126. - PubMed
- Bustamante, C., J. F. Marko, E. D. Siggia, and S. Smith. 1994. Entropic elasticity of λ-phage DNA. Science. 265:1599–1600. - PubMed
- Perkins, T. T., S. R. Quake, D. E. Smith, and S. Chu. 1994. Relaxation of a single DNA molecule observed by optical microscopy. Science. 264:822. - PubMed
- Marko, J. F., and E. D. Siggia. 1995. Stretching DNA. Macromolecules. 28:8759–8770.
Publication types
MeSH terms
Substances
LinkOut - more resources
Full Text Sources
Miscellaneous